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The lines joining the origin to the point of intersection of \[3{{x}^{2}}+mxy-4x+1=0\]and $2x+y-1=0$ are at right angles. Then which of the following is/are possible value/s of \[m\]?
This question has multiple correct options.
a) \[-4\]
b) \[4\]
c) \[7\]
d) \[3\]

Last updated date: 13th Jun 2024
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Hint: Here we use the application of Homogenization of equations and solve the equations of lines. Then we will find the value of m.Homogenization means it should make the degree of every term the same.

Complete step-by- step solution:
Given: Lines are \[3{{x}^{2}}+mxy-4x+1=0\] and$2x+y-1=0$intersect at the right angle joining the origin to the point of intersection.
Given line is $2x+y-1=0....................(1)$
And, curve is \[3{{x}^{2}}+mxy-4x+1=0...................(2)\]
Now, By the application of Homogenising eq (2) with the help of eq (1):
Then, \[\]$3{{x}^{2}}+mxy-4x(2x+y)+{{(2x+y)}^{2}}=0$
Simplify the above equation, taking like terms together and applying additive and subtractive property wherever applicable
$\Rightarrow$ $3{{x}^{2}}+mxy-8{{x}^{2}}-4xy+4{{x}^{2}}+{{y}^{2}}+4xy$
$\Rightarrow$ $-{{x}^{2}}+mxy+{{y}^{2}}=0$
Hence the equations of the lines are given by ${{x}^{2}}-mxy-{{y}^{2}}=0$
Hence the lines are (perpendicular for all values of m) at the right angles as $a+b=0$, when ${{h}^{2}}\ge ab$
i.e. $\dfrac{{{m}^{2}}}{4}+1\ge 0$
Simplify the above condition -
  & \Rightarrow {{m}^{2}}+4\ge 0 \\
 & \Rightarrow {{m}^{2}}\le -4 \\
 & \Rightarrow m\le \mp 4 \\
Therefore, the possible values of m is $\pm 4$

Hence, from the given multiple options, the correct option is A and B.

Note: In these types of problems we use application of homogenization of equations. Homogenization is the process of making equations homogeneous. Which means to make the degree of every term the same. As after homogenization both the equations will be quadratic and now, we will be able to solve the equation and get the value of m. Also, we should be very careful about inequality as when sign changes sign of inequality also changes. Like here $\Rightarrow {{m}^{2}}+4\ge 0\Rightarrow {{m}^{2}}\le -4\Rightarrow m\le \mp 4$ , as${{m}^{2}}+4\ge 0$ and ${{m}^{2}}\le -4$.